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Re: Globally limiting precision or accuracy

• To: mathgroup at smc.vnet.net
• Subject: [mg61032] Re: [mg61010] Globally limiting precision or accuracy
• From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
• Date: Sat, 8 Oct 2005 02:48:40 -0400 (EDT)
• References: <200510070737.DAA03251@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

On 7 Oct 2005, at 16:37, Lee Newman wrote:

> Dear Group,
>
> Situation
> -  I have a large neural network model (my own code, not Wolfram's
> toolbox) that currently has a run time of about 20hours.
> -  I am in the process of trying to profile and optimize the code
> (mostly matrix computations) to reduce the run time.
> -  All of the computationas that I do are numerical.
>
> Questions
> (1) If I am not concerned about numerical accuracy beyond 3 decimal
> places for any of the computations in the model, can I improve
> performance by telling mathematica to globally restrict its  
> accuracy (or
> precision) for all computations?
>
> (2) If so, how do I do this?  Is it as simple as setting
> $MachinePrecision=3?

I don't think this can be done at all. You can of course Unprotect 
[$MachinePrecision] and set the global $MachinePrecision variable to  
3, but this won't make any difference at all to machine precision  
calculations because (as the name tells you) that is decided by the  
"machine" i.e. your processor.
I am pretty sure there is no way to improve on the performance of  
machine precision floating point computations by any Mathematica  
commands. In fact, I think, the only way to do that is to get a  
faster computer.
On the other hand you should be aware that when you use machine  
precision arithmetic Mathematica does not attempt to keep any track  
of the precision of your computations. If you come across an ill- 
conditioned expression it will return nonsensical answers without any  
warning. You certainly cannot be sure that any machine precision  
computation is accurate even to three digits. To be sure of that you  
have to work with extended precision numbers, because only then  
Mathematica will keep track of precision. By using exact quantities  
as input in your expression and computing N[expr,3] you should  
normally get guaranteed three digits of precision in a relatively  
efficient way. E.g.


N[Pi+23/234,3]


3.24


Precision[%]


3.

This will always be slower than when you do this with  
MachinePrecision (which formally gives more digits of "precision" but  
without any guarantee.



> Is there a global way (rather than local use of
> N[]) to ensure that all computations are done numerically, and with
> machine precision?
>

As long as the numbers in your input are machine precision numbers,  
e.g. 1.3 or 2.3456 etc, all computations will be done with machine  
precision, with all that it entails (including the possibility of  
nonsensical answers). In fact, usually it is enough that you  
expression involves a single machine precision number for the entire  
expression to be computed using machine precision as in


1.2*Pi+123


126.77


Precision[%]


MachinePrecision

Andrzej Kozlowski

• Follow-Ups:
  • Re: Re: Globally limiting precision or accuracy
    • From: Andrzej Kozlowski <akoz@mimuw.edu.pl>

• References:
  • Globally limiting precision or accuracy
    • From: Lee Newman <leenewm@umich.edu>